American Institute of Mathematical Sciences

Advanced
May  2016, 15(3): 965-989. doi: 10.3934/cpaa.2016.15.965

Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials

Weiwei Ao 1, , Liping Wang 2, and  Wei Yao 3,

1. 

Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2
2. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
3. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Chile

Received  October 2015 Revised  November 2015 Published  February 2016

Without any symmetric conditions on potentials, we proved the following nonlinear Schrödinger system \begin{eqnarray} \left\{\begin{array}{ll} \Delta u-P(x)u+\mu_1u^3+\beta uv^2=0, \quad &\mbox{in} \quad R^2\\ \Delta v-Q(x)v+\mu_2v^3+\beta vu^2=0, \quad &\mbox{in} \quad R^2 \end{array} \right. \end{eqnarray} has infinitely many non-radial solutions with suitable decaying rate at infinity of potentials $P(x)$ and $Q(x)$. This is the continued work of [8]. Especially when $P(x)$ and $Q(x)$ are symmetric, this result has been proved in [18].
Keywords: intermediate reduction method., non-symmetric potentials, infinitely many solutions, Schrödinger system.
Mathematics Subject Classification: Primary: 35B35, 35J25; Secondary: 35B40, 92D2.
Citation: Weiwei Ao, Liping Wang, Wei Yao. Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials. Communications on Pure & Applied Analysis, 2016, 15 (3) : 965-989. doi: 10.3934/cpaa.2016.15.965
References:
[1]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).   Google Scholar

[5]

K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.  doi: 10.1016/S0375-9601(01)00369-3.  Google Scholar

[6]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[7]

N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[8]

M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.  doi: 10.1007/s00526-014-0756-3.  Google Scholar

[9]

Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().   Google Scholar

[10]

F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.   Google Scholar

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[12]

T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.  doi: 10.1016/j.physd.2006.07.009.  Google Scholar

[13]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[14]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.   Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.  doi: 10.1038/43136.  Google Scholar

[16]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[17]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.   Google Scholar

[18]

S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[19]

E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718.   Google Scholar

[20]

S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[21]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[22]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.  doi: 10.1112/plms/pdq051.  Google Scholar

[23]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.   Google Scholar

[24]

J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.   Google Scholar

[25]

J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[26]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[27]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.   Google Scholar

show all references

References:
[1]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[2]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[3]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[4]

J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).   Google Scholar

[5]

K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.  doi: 10.1016/S0375-9601(01)00369-3.  Google Scholar

[6]

M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[7]

N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.  doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[8]

M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.  doi: 10.1007/s00526-014-0756-3.  Google Scholar

[9]

Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., ().   Google Scholar

[10]

F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.   Google Scholar

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[12]

T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.  doi: 10.1016/j.physd.2006.07.009.  Google Scholar

[13]

Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.  doi: 10.1007/s00220-008-0546-x.  Google Scholar

[14]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.   Google Scholar

[15]

M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.  doi: 10.1038/43136.  Google Scholar

[16]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[17]

B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.   Google Scholar

[18]

S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.  doi: 10.1007/s00205-012-0598-0.  Google Scholar

[19]

E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718.   Google Scholar

[20]

S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.  doi: 10.1007/s00205-008-0172-y.  Google Scholar

[21]

L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.  doi: 10.1090/S0002-9947-10-04955-X.  Google Scholar

[22]

L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.  doi: 10.1112/plms/pdq051.  Google Scholar

[23]

L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.   Google Scholar

[24]

J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.   Google Scholar

[25]

J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.  doi: 10.1007/s00205-008-0121-9.  Google Scholar

[26]

J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.  doi: 10.1007/s00526-009-0270-1.  Google Scholar

[27]

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.   Google Scholar

[1]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021010
[2]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[3]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298
[4]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011
[5]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[6]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392
[7]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294
[8]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021008
[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[10]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323
[11]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[12]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[14]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[15]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292
[16]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018
[17]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115
[18]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[19]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
[20]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences